Cubic pencils of lines and bivariate interpolation

AbstractCubic pencils of lines are classified up to projectivities. Explicit formulae for the addition of lines on the set of nonsingular lines of the pencils are given. These formulae can be used for constructing planar generalized principal lattices, which are sets of points giving rise to simple Lagrange formulae in bivariate interpolation. Special attention is paid to the irreducible nonsingular case, where elliptic functions are used in order to express the addition in a natural form

A Geometric Interpretation and Comparison of the Methods Of Ordinary Least Square (OLS) and Bivariate Lagrange Interpolation

In this study, the consumer prices, real gross domestic product (GDP) and unemployment for Germany and Turkey between 2006 and 2011 are geometrically interpreted by using Lagrange interpolation and OLS method. The coefficients of linear regression models are obtained by matrix display of OLS method. The Lagrange interpolation polynomial is considered in bivariate situation and we aim a different formulation. Thanks to the considered methods, it will be possible to have an idea about the unemployment rate status in Germany and Turkey. We use two different methods for the prediction of unemployment rates, which are developed by different equations in order to predict third variable by using two variables. Our main purpose is to determine whether an equation gives the correct guess rather than numerical expression. Besides, we have tried to state geometric display to data. Keywords: Lagrange interpolation, ordinary least square (OLS), regression, geometrical display, matrix display

Observations on interpolation by total degree polynomials in two variables

In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon–Berzolari and Chung–Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal that allow us to draw conclusions on the geometry of the point configuration

Observations on interpolation by total degree polynomials in two variables

In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon–Berzolari and Chung–Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal that allow us to draw conclusions on the geometry of the point configuration